Method of collective fabrication of calibration-free temperature and/or strain sensors by matching of resonators on the basis of resonant frequency and static capacitance criteria

ABSTRACT

A method of collective fabrication of remotely interrogatable sensors, wherein the method may include fabricating fabricating a first series of first resonators exhibiting a first resonant frequency at ambient temperature and a first static capacitance and fabricating a second series of second resonators exhibiting a second resonant frequency at ambient temperature and a second static capacitance. The method may also include performing a series of electrical measurements of the set of the first series of first resonators and of the set of the second series of second resonators, so as to determine first pairs and second pairs of resonant frequency and of capacitance of each of the first and second resonators and performing a series of matching of a first resonator and of a second resonator.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to foreign Patent Application FR 0902993, filed on Jun. 19, 2009, the disclosure of which is incorporatedherein by reference in its entirety.

FIELD OF THE INVENTION

The field of the invention is that of passive surface acoustic wavesensors, also known as “SAW” sensors, making it possible to perform forexample measurements of temperature and/or of pressure/stressesremotely, and more precisely that of the collective fabrication of suchsensors.

BACKGROUND OF THE INVENTION

One type of temperature sensor can typically consist of two SAWresonators denoted R₁ and R₂ and undertake differential measurements.For this purpose the two resonators are designed to have differentresonant frequencies.

Typically, each resonator is composed of a transducer withinter-digitated combs, consisting of an alternation of electrodes, whichare repeated with a certain periodicity called the metallization period,deposited on a piezoelectric substrate that may advantageously bequartz. The electrodes, advantageously aluminium or aluminium alloy(produced by a photolithography method), exhibit a low thicknessrelative to the metallization period (typically, a few hundrednanometers to a few micrometers). For example for a sensor operating at433 MHz, the metal (aluminium for example) thickness used can be of theorder of 100 to 300 nanometers, the metallization period and theelectrode width possibly being respectively of the order of 3.5 μm and2.5 μm.

One of the ports of the transducer is for example linked to the livepoint of a Radio Frequency (RF) antenna and the other to earth or elsethe two ports are linked to the antenna if the latter is symmetric(dipole for example). The field lines thus created between twoelectrodes of different polarities give rise to surface acoustic wavesin the zone of overlap of the electrodes.

The transducer is a bi-directional structure, that is to say the energyradiated towards the right and the energy radiated towards the left havethe same intensity. By arranging electrodes on either side of thetransducer, the said electrodes playing the role of reflector, aresonator is produced, each reflector partially reflecting the energyemitted by the transducer.

If the number of reflectors is multiplied, a resonant cavity is created,characterized by a certain resonant frequency. This frequency dependsfirstly on the speed of propagation of the waves under the network, thesaid speed depending mainly on the physical state of the substrate, andtherefore sensitive for example to temperature. In this case, this isthe parameter which is measured by the interrogation system and it is onthe basis of this measurement that a temperature can be calculated.

It is recalled that the variation of the resonant frequency as afunction of temperature of a quartz resonator is determined by thefollowing formula:f(T)=f ₀[1+CTF ₁(T−T ₀)+CTF ₂(T−T ₀)²]

-   -   With f₀ the frequency at T₀, T₀ being the reference temperature        (25° C. by convention), CTF₁ the first-order coefficient (ppm/°        C.) and CTF₂ the second-order coefficient (ppb/° C.²).

The two resonators can use different wave propagation directions,produced though an inclination of the different inter-digitatedelectrode combs on one and the same substrate, for example quartz.

The two resonators can also advantageously use different quartz cutsmaking it possible to endow them with different resonant frequencies, inthis instance for the resonator R₁ the quartz cut (YX1)/θ₁ and for theresonator R₂: the cut (YX1)/θ₂ with reference to the IEEE standardexplained hereinafter, the two resonators using propagation which iscollinear with the crystallographic axis X.

Whatever solution is adopted for creating different resonantfrequencies, the fact of using a differential structure presents severaladvantages. The first is that the frequency difference of the resonatorsis almost linear as a function of temperature and the residualnon-linearities taken into account by the calibration of the sensor.Another advantage of the differential structure resides in the fact thatit is possible to sidestep the major part of the ageing effects.

It is recalled that the expression “calibration operation” denotes thedetermination of so-called calibration parameters A₀, A₁ and A₂ of thefollowing function:T=A ₀±√{square root over (A ₁ +A ₂ Δf)}

When these parameters are defined, a differential measurement offrequency then makes it possible to determine a temperature.

Generally, resonators are produced collectively on wafers 100 mm indiameter, typically this might involve fabricating about 1000 specimenson one and the same wafer. This therefore gives 1000 specimens ofresonators R₁ and 1000 specimens of resonators R₂, each temperaturesensor comprising a pair of resonators R₁ and R₂.

The calibration operation is nonetheless expensive in terms of timesince it makes it necessary to measure for each sensor the frequencydifference between the two resonators at three different temperatures atthe minimum and moreover requires the serialization of each sensor(corresponding to the identification of a sensor—calibrationcoefficients pair for each sensor).

It is for example possible to envisage storing the calibrationcoefficients A₀, A₁, A₂ in the interrogation system. This configurationrequires, in the event of a change of sensor, that the new coefficientsbe stored in the interrogation system.

One of the aims sought in the present invention is to produce acalibration-free temperature sensor while retaining good precision inthe temperature measurement.

For this purpose it is necessary to control on the one hand thedispersion in the difference in resonant frequencies of the resonatorsR₁ and R₂, and on the other hand the dispersion in the coefficients CTF₁and CTF₂ (first-order and second-order temperature coefficients), or atleast the difference in these coefficients CTF₁ and CTF₂ when adifferential measurement is carried out, as is demonstrated hereinafterand by virtue of the following various reminders:

1) Concerning Crystalline Orientation

In order to define the crystalline orientations, the IEEE standard isused. This designation uses the following 2 reference frames:

-   -   the crystallographic reference frame (X, Y, Z).    -   the working reference frame (w, l, t) defined by the surface of        the substrate (normal to {right arrow over (t)}) and the        direction of propagation of the surface waves (axis {right arrow        over (l)}).

The designation of a cut is of the type (YX wlt)/φ/θ/ψ with:

-   -   YX two crystalline axes making it possible to place the working        reference frame with respect to the crystallographic reference        frame before any rotation. The first axis is along the axis t,        normal to the surface whereas the second is along the axis l.        The third axis of the working reference frame w is given by the        sense of the right-handed trihedron (w, l, t).    -   w, l, t indicates a series of axes around which it is possible        to perform successive rotations by respective angles φ, θ, ψ. In        the subsequent description, the variables φ, θ, ψ are associated        with rotations around the respective axes w, l, t.

2) Concerning the Geometry of the Saw Resonator:

The dimensions characterizing a surface wave device consisting ofinter-digitated electrode combs Ei, which are symmetric with respect toan axis Ac and deposited on the surface of a piezoelectric substrate aredenoted in the following manner and illustrated in FIG. 1:

the metallization period denoted: “p”;

the wavelength denoted: “λ”, with λ=2·p;

the electrode width denoted: “a”;

the metallization thickness denoted “h”.

In general, to sidestep the operating frequency of the device, thefollowing normalized variables are actually used:

-   -   the metallization ratio a/p, ratio of electrode width to the        metallization period;    -   the normalized metallization thickness h/λ ratio of the        metallization thickness to the wavelength λ=2·p.

3) Concerning the Laws of Variations with Temperature of 2 Surface WaveResonators:

As defined previously it is possible to express the frequency behavioursof the two resonators respectively by the following equations:For the resonator R ₁ : f ₁(T)=f ₀₁·(1+C ₁₁·(T−T ₀)+C ₂₁·(T−T ₀)²)  (1)

With: f₁(T) the resonant frequency of R₁ as a function of temperature

f₀₁ the resonant frequency of R₁ at the temperature T₀ (generally 25°C.);

C₁₁ the 1^(st)-order temperature coefficient (generally called CTF1) ofR₁;

C₂₁ the 2^(nd)-order temperature coefficient (generally called CTF2) ofR₁;For the resonator R ₂ : f ₂(T)=f ₀₂·(1+C ₁₂·(T−T ₀)+C ₂₂·(T−T ₀)²)  (2)

With: f₂(T) the resonant frequency of R₂ as a function of temperature

f₀₂ the resonant frequency of R₂ at the temperature T₀ (generally 25°C.);

C₁₂ the 1^(st)-order temperature coefficient (generally called CTF1) ofR₂;

C₂₂ the 2^(nd)-order temperature coefficient (generally called CTF2) ofR₂;

In the general case, the resonant frequency at 25° C. and the1^(st)-order and 2^(nd)-order temperature coefficients depend mainly:

on the chosen crystalline orientation;

on the metallization period of “p” for f₀ alone;

on the normalized metallization thickness h/λ;

on the metallization ratio a/p.

And generally, the frequency difference is a function of temperaturewhich can therefore be expressed in the following manner:

$\begin{matrix}\begin{matrix}{{\Delta\;{f(T)}} = {{f_{2}(T)} - {f_{1}(T)}}} \\{= {f_{02} - f_{01} + {\begin{pmatrix}{{C_{12} \cdot f_{02}} -} \\{C_{11} \cdot f_{01}}\end{pmatrix} \cdot \left( {T - T_{0}} \right)} +}} \\{\begin{pmatrix}{{C_{22} \cdot f_{02}} -} \\{C_{21} \cdot f_{01}}\end{pmatrix} \cdot \left( {T - T_{0}} \right)^{2}} \\{= {\Delta_{0} + {s \cdot \left( {T - T_{0}} \right)} + {ɛ \cdot \left( {T - T_{0}} \right)^{2}}}}\end{matrix} & (3)\end{matrix}$

With: Δ₀=f₀₂−f₀₁ the difference in resonant frequency at the temperatureT₀;

s=C₁₂·f₀₂−C₁₁·f₀₁ the 1^(st)-order differential coefficient

ε=C₂₂·f₀₂−C₂₁·f₀₁ the 2^(nd)-order differential coefficient

The calibration coefficients make it possible on the basis of ameasurement of the frequency difference to get back to the temperatureinformation. It can be shown that:

$\begin{matrix}{T = {{T_{0} + \frac{{- s} \pm \sqrt{s^{2} - {4{ɛ\left( {\Delta_{0} - {\Delta\; f}} \right)}}}}{2ɛ}} = {A_{0} \pm \sqrt{A_{1} + {A_{2}\Delta\; f}}}}} & (4)\end{matrix}$

Where A₀, A₁ and A₂ are the calibration coefficients as explained in thepreamble of the present description.

4) Concerning Manufacturing Dispersions:

The methods of fabrication of resonators being controlled with a certainprecision, the crystalline orientation (φ, θ, ψ) and the geometry of theresonator (related to the parameters a and h alone, in effect it isconsidered that the metallization period p is perfectly controlled) arenever, in practice, exactly those aimed at and moreover they are notperfectly reproducible.

For a sufficiently large sample, these parameters follow Gaussiandistributions (law of large numbers) whose means and standard deviationscan be determined experimentally. The whole set of variations of thefive parameters φ, θ, ψ, a and h is called manufacturing dispersions.

The parameters f₀, C₁₁, C₁₂ and C₂₁, C₂₂ being dependent on φ, θ, ψ, aand h, can also be controlled with a certain precision and can followdistributions centred around a mean with a certain standard deviation.

The applicant has started from the assumption that there were threepredominant parameters in terms of manufacturing dispersions withrespect to the set of five parameters f₀, C₁₁, C₁₂ and C₂₁, C₂₂.

The three predominant parameters in the manufacturing dispersions arethe following:

-   -   the dispersion in the angle of cut θ which corresponds in IEEE        notation to the cut (YX1)/θ;    -   the dispersion in the metallization thickness a;    -   the dispersion in the electrode width h.

Indeed, the cuts of the substrates are chosen such that they comply withthe criteria: φ=0 and ψ=0 thereby corresponding to the crystallineorientation (YXwlt)/φ=0/ψ=0 in IEEE notation.

Now, the points φ=0 and ψ=0 correspond to points at which all thederivatives with respect to φ and ψ vanish. The variations of thefollowing parameters taken into account (f₀, C₁, C₂) can be consideredzero around these points:

$\begin{matrix}{{{{{{{{{{{{{{{{{\frac{\partial f_{0}}{\partial\varphi}}_{\varphi = 0} = 0}\frac{\partial C_{1}}{\partial\varphi}}}_{\varphi = 0} = 0}\frac{\partial C_{2}}{\partial\varphi}}}_{\varphi = 0} = 0}\frac{\partial f_{0}}{\partial\psi}}}_{\psi = 0} = 0}\frac{\partial C_{1}}{\partial\psi}}}_{\psi = 0} = 0}\frac{\partial C_{2}}{\partial\psi}}}_{\psi = 0} = 0} & (5)\end{matrix}$

Typically and by way of example, the following dispersions in these 3parameters can be considered:

a dispersion in electrode width: Δa=+/−0.06 μm;

a dispersion in metallization thickness: Δh=+/−30 Angströms;

a dispersion in angle of cut: Δθ=+/−0.05°.

Assuming the 3 parameters follow Gaussian distributions, +/−3 times thestandard deviation of the relevant parameter is called the dispersion:

Δa=+/−3·σ(a)

Δh=+/−3·σ(h)

Δθ=+/−3·σ(θ)

With σ(a), σ(h), σ(θ) respectively the standard deviations of theelectrode width a, of the metallization thickness h and of the angle ofcut θ.

Note that for a Gaussian distribution with mean μ and standard deviationσ, 99.74% of the most probable population is in the interval [μ−3·σ,μ+3·σ]:P(μ−3·σ<X<μ+3·σ)=0.9974  (6)

In the subsequent description, the expression “nominal value” refers tothe values of the parameter a, h or θ aimed at during fabrication andcalled hereinafter: a_(nom), h_(nom), θ_(nom).

Moreover, for each of the 3 parameters, the following cases areconsidered:a _(min) =a _(nom) −Δa a _(max) =a _(nom) +Δah _(min) =h _(nom) −Δh h _(max) =h _(nom) +Δhθ_(min)=θ_(nom)−Δθ θ_(max)=θ_(nom)+Δθ  (7)

5) Concerning the Sensor Calibration Operation:

The parameters f₀, C₁, C₂ controlled with a certain precision, aredistributed according to a distribution centred around a mean with acertain standard deviation. The laws of variations with temperature ofthe resonators are therefore not identical for all the sensors and thesame holds for the calibration coefficients.

To obtain maximum precision of temperature measurement, the calibrationcoefficients must therefore be calculated individually for each sensor.For this purpose, it is necessary to measure Δf(T) over the whole of thetemperature span where the sensor is used so as to fit the coefficientsΔ₀, s, ε and ultimately calculate A₀, A₁ and A₂.

This operation is very lengthy and hardly compatible with high-volumeproduction, one seeks therefore to sidestep it.

Among the solutions that may be conceived for accomplishing collectivefabrication of calibration-free SAW sensors it is conceivable to use asuite of common calibration coefficients for a set of sensors whilemaintaining acceptable measurement precision. Moreover, a limited numberof sensors can be measured temperature-wise (representative sample)making it possible to determine a mean calibration coefficients suiteused for the whole set of sensors. It is then advisable that a suite ofcalibration coefficients should be common to the largest possible numberof sensors, the ideal even being that a suite of coefficients should becommon to all the sensors of a given type (defined by the crystallineorientation and the geometry of each of the 2 resonators). Thistherefore produces what is called a “calibration-free sensor”.

Generally, by considering the law of differential variations withtemperature, given by expression (3), it is seen that it is necessary toreduce the dispersions in Δ₀, s and ε, if one wishes to have a suite ofcommon calibration coefficients for all the sensors, while having goodprecision of frequency measurement.

One solution is to reduce the dispersions in f₀₁, C₁₁, C₂₁, f₀₂, C₁₂ andC₂₂. This leads to carrying out a sorting operation on each of the 3parameters of the two resonators. This approach is, however, not thatadopted in the present invention for the following reasons:

-   -   one of the objectives is to not measure the sensors        temperature-wise individually, therefore the values of C₁₁, C₂₁,        C₁₂ and C₂₂ are not known for each sensor.    -   moreover, calculations have shown that a sorting operation such        as presented reduces the yields too much if acceptable        measurement precision is desired.

SUMMARY OF THE INVENTION

In this context and to solve the aforementioned problems, the presentinvention relates to a novel method of collective fabrication ofcalibration-free sensors making it possible to retain acceptablemeasurement precision.

More precisely, one embodiment of the present invention provides amethod of collective fabrication of remotely interrogatable sensors,each sensor comprising at least one first resonator and one secondresonator, each resonator comprising acoustic wave transducers designedsuch that they exhibit respectively a first and a second operatingfrequency, in which the method comprises:

-   -   the fabrication of a first series of first resonators RT_(1i)        exhibiting a first resonant frequency at ambient temperature        f_(1i) and a first static capacitance C_(1i);    -   the fabrication of a second series of second resonators RT_(2j)        exhibiting a second resonant frequency at ambient temperature f₂        and a second static capacitance C_(2j);    -   a series of electrical measurements of the set of the first        series of first resonators and of the set of the second series        of second resonators, so as to determine first pairs (f_(1i),        C_(1i)) and second pairs (f_(2j), C_(2j)) of resonant frequency        and of static capacitance of each of the first and second        resonators; and    -   a series of matchings of a first resonator RT_(1i) and of a        second resonator RT_(2j) according to the aggregate of the        following two criteria: the dispersion in the difference in        resonant frequency (f_(1i)−f_(2j)) is less than a first        threshold value (Sf) and the dispersion in the difference in        static capacitance (C_(1i)−C_(2j)) is less than a second        threshold value of (Sc).

According to a variant of the invention, the electrical measurements areperformed by determining measurements of the reflection coefficient S11or measurements of admittance Y11 or else measurements of impedance Z11.

According to a variant of the invention the electrical measurements areperformed with a network analyzer.

According to a variant of the invention, the measurements of staticcapacitance are carried out with a high-precision capacimeter.

According to a variant of the invention, the first and second resonantfrequencies are similar and situated in the ISM frequency span (433.05MHz, 434.79 MHz), the threshold value Sf being less than or equal toabout a few kHz and/or the threshold value Sc being less than or of theorder of a femtoFarad.

According to a variant of the invention, the method comprises for eachfirst resonator of the first series, the selection of a second resonatorof the second series so as to satisfy the two matching criteria.

According to a variant of the invention, the method comprises thefabrication of first resonators on a first substrate and the fabricationof second resonators on a second substrate.

According to a variant of the invention, the resonators are produced onquartz substrates of different cuts.

According to a variant of the invention, the first and second substratesare defined by angles of cut θ according to the IEEE standard (YX1)/θ,of 24° and 34° so as to generate resonators of frequency 433 MHz and 434MHz.

According to a variant of the invention, the method further comprises:

-   -   the fabrication of first resonators (RT_(1i)) on a first        substrate (S1) and the fabrication of second resonators        (RT_(i2)) on a second substrate (S2);    -   unit slicings of first and of second chips comprising        respectively the first and second resonators from the said        substrates;    -   the matching of a first and of a second chip;    -   the assembling of the pairs of chips in a package.

According to a variant of the invention, the method further comprises:

-   -   the fabrication of first resonators on a first substrate and the        fabrication of second resonators on a second substrate;    -   unit slicings of first and of second chips comprising        respectively the first and second resonators from the said        substrates;    -   the individual packaging of the first chips and of the second        chips in individual packages;    -   the matching of a first and of a second previously packaged        chip.

According to a variant of the invention, the sensor is a temperaturesensor.

According to a variant of the invention, the first resonators areoriented on the first substrate in a first direction, the secondresonators are oriented on the second substrate in a second direction,the said directions corresponding to the directions of propagation ofthe surface waves, and in such a way that the first direction makes anon-zero angle with the second direction.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be better understood and other advantages will becomeapparent on reading the description which follows given by way ofnon-limiting example and by virtue of the appended figures among which:

FIG. 1 illustrates a resonator structure according to an embodiment ofthe present invention;

FIG. 2 illustrates the evolution of the resonant frequency (MHz) as afunction of h/λ in % for an angle of cut of 24°, for various pairs (θ,a);

FIG. 3 illustrates the evolution of the resonant frequency (MHz) as afunction of h/λ in % for an angle of cut of 34°, for various pairs (θ,a);

FIG. 4 illustrates the evolution of the static capacitance (pF) as afunction of the metallization ratio for an angle of cut of 24°, forvarious pairs (θ, h);

FIG. 5 illustrates the evolution of the static capacitance (pF) as afunction of the metallization ratio for an angle of cut of 34°, forvarious pairs (θ, h);

FIG. 6 illustrates the error probability density obtained during amatching operation with a criterion +/−0.2 σ(X) used in a method ofcollective fabrication according to an embodiment of the presentinvention;

FIG. 7 the error probability density obtained during a matchingoperation with a criterion +/−0.1 σ(X) used in a method of collectivefabrication according to an embodiment of the present invention.

DETAILED DESCRIPTION

A method of collective fabrication of remotely interrogatable passiveacoustic wave sensors advantageously produces at least two resonators,arising from the fabrication of two series of resonators, matchedpairwise.

Various embodiments of the present invention are described hereinafterwithin the framework of two resonators exhibiting similar resonantfrequencies, typically this is the case with a frequency f₀₁˜433.6 MHzand a frequency f₀₂˜434.4 MHz.

The resonators R1 can be produced on the surface of an (XY1)/24 quartzcut and the resonators R2 can be produced on the surface of an (XY1)/34quartz cut.

Nonetheless, the invention could be implemented with other cuts.

The applicant has started from the finding that it was possible toeffect the following approximation: f₀₂≈f₀₁ and df₀₂≈df₀₁.

Typically this approximation can be made when (f₀₂−f₀₁)/f₀₁<<1, this istypically the case when there are two orders of magnitude of difference.

By way of example with a frequency f₀₁˜433.6 MHz and a frequencyf₀₂˜434.4 MHz and 3·σ(f₀₂)≈3·σ(f₀₁)=110 kHz, the approximation isacceptable.

The differential coefficients then become:s=C ₁₂ ·f ₀₂ −C ₁₁ ·f ₀₁ ≈f ₀₁·(C ₁₂ −C ₁₁)ε=C ₂₂ ·f ₀₂ −C ₂₁ ·f ₀₁ ≈f ₀₁·(C ₂₂ −C ₂₁)

And the dispersions corresponding to the partial derivatives can bewritten:ds=df ₀₁·(C ₁₂ −C ₁₁)+f ₀₁ ·d(C ₁₂ −C ₁₁)dε=df ₀₁·(C ₂₂ −C ₂₁)+(C ₂₂ −C ₂₁)

By way of example let us consider that the resonator R₁ uses the quartzcut (YX1)/24 and the resonator R₂ the cut (YX1)/34. These two resonatorscan potentially be used for a differential measurement of thetemperature in a span of [−20, 160]° C. and using the ISM band [433.05,434.79] MHz.

Under these conditions we have:

C₁₁=6.8 ppm/° C.

C₂₁=−30.7 ppb/° C.²

C₁₂=0.4 ppm/° C.

C₂₂=−38.1 ppb/° C.²

f₀₁˜433.6 MHz

Δf₀₁≈Δf₀₂=3·σ(f₀₁)=110 kHz

Δ(C₁₂−C₁₁)=3·σ(C₁₂−C₁₁)=0.456 ppm/° C.

Δ(C₂₂−C₂₁)=3·σ(C₂₂−C₂₁)=0.41 ppb/° C.²

Hence:

$\begin{matrix}{{\Delta\; s} = {{\Delta\;{f_{01} \cdot {{C_{12} - C_{11}}}}} + {f_{01} \cdot {\Delta\left( {C_{12} - C_{11}} \right)}}}} \\{= {{110*10^{3}*6.4*10^{- 6}} + {433.6*10^{6}*0.456*10^{- 6}}}} \\{= {0.704 + 197.7216}}\end{matrix}$

It is thus seen that Δf₀₁·|C₁₂−C₁₁|<<f₀₁·Δ(C₁₂−C₁₁)

It is therefore possible to make the approximation Δs≈f_(0l)·Δ(C₁₂−C₁₁)

Likewise:

$\begin{matrix}{{\Delta\; ɛ} = {{\Delta\;{f_{01} \cdot {{C_{22} - C_{21}}}}} + {f_{01} \cdot {\Delta\left( {C_{22} - C_{21}} \right)}}}} \\{= {{110*10^{3}*7.4*10^{- 9}} + {433.6*10^{6}*0.41*10^{- 9}}}} \\{= {{814*10^{- 6}} + {177.776*10^{- 3}}}}\end{matrix}$

It is thus seen that: Δf₀₁·|C₂₂−C₂₁|<<f₀₁·Δ(C₂₂−C₂₁)

It is therefore possible to make the approximation: Δε≈f₀₁·Δ(C₂₂−C₂₁)

Returning to the 3 differential temperature coefficients, theirdispersions can therefore be written:−dΔ ₀ =d(f ₀₂ −f ₀₁)−ds≈f ₀₁ ·d(C ₁₂ −C ₁₁)−dε≈f ₀₁ ·d(C ₂₂ −C ₂₁)  (8)

This result can be extended to cuts other than those cited above sincethe orders of magnitude remain the same whatever the cut.

The applicant has shown that the dispersion in the sensor temperaturelaws depends essentially on the dispersion in the frequency differencewhich has formed the subject of a patent application filed by theapplicant and published under the reference FR 2 907 284, and thedispersions in the differences of CTFs between the 2 resonators.

It is therefore possible to reduce the dispersion in the sensortemperature laws by carrying out a matching of the 2 resonators. That isto say by selecting from among the sets of specimens of resonators R₁and R₂ pairs of specimens such that:(f ₀₂ −f ₀₁)−ξ(f ₀₂ −f ₀₁)<(f ₀₂ −f ₀₁)<(f ₀₂ −f ₀₁)+ξ(f ₀₂ −f ₀₁)(C ₁₂ −C ₁₁)−ξ(C ₁₂ −C ₁₁)<(C ₁₂ −C ₁₁)<(C ₁₂ −C ₁₁)+ξ(C ₁₂ −C ₁₁)(C ₂₂ −C ₂₁)−ξ(C ₂₂ −C ₂₁)<(C ₂₂ −C ₂₁)<(C ₂₂ −C ₂₁)+ξ(C ₂₂ −C ₂₁)  (9)

With ξ the permitted variation in the difference considered.

For example, for the cuts considered, it is possible to carry out amatching satisfying:795 kHz<(f ₀₂ −f ₀₁)<805 kHz with ξ(f ₀₂ −f ₀₁)=5 kHz−6.45 ppm/° C.<(C ₁₂ −C ₁₁)<−6.35 ppm/° C. with ξ(C ₁₂ −C ₁₁)=0.05 ppm/°C.7.35 ppb/° C.²<(C ₂₂ −C ₂₁)<7.45 ppb/° C.² with ξ(C ₂₂ −C ₂₁)=0.05 ppb/°C.²

The advantage of matching is to allow much higher yields than a sortingoperation on the parameters of resonators taken separately for identicaltemperature law dispersions.

It is thus apparent that the matching can reduce the temperature lawdispersions while maintaining acceptable yields.

It is explained hereinafter how it is thus possible to carry out amatching based on the difference of CTFs without individually measuringthe resonators temperature-wise, this constituting a majorcharacteristic of the present invention.

The applicant has started from the finding that the resonators generallyuse points said to have insensitivity to the width of electrodes so thatthe resonant frequency is “almost” independent of the latter by virtueof imposed design rules. For this purpose, a point is sought for which:

$\begin{matrix}{{\frac{\partial f_{0}}{\partial a}}_{a = a_{nom}} = 0} & (10)\end{matrix}$

The resonant frequency of the resonator then depends only on themetallization thickness and the angle of cut θ.

Moreover, it may easily be shown that the dispersions in resonantfrequencies depend very significantly on the dispersions inmetallization thickness h.

FIG. 2 illustrates this effect for an exemplary cut with θ=24°. It isapparent that the curves corresponding to variations Δθ of + or −0.05°around 24° all coincide for various values of a (a_(nom), a_(min) anda_(max)), the set of curves being relative to the following pairs:(θ_(min), a_(min)), (θ_(min), a_(nom)), (θ_(min), a_(max)), (θ_(nom),a_(min)), (θ_(nom), a_(nom)), (θ_(nom), a_(max)), (θ_(max), a_(min)),(θ_(max), a_(nom)), (θ_(max), a_(max)).

The same phenomenon is obtained with an angle of cut θ=34° andillustrated by FIG. 3.

It emerges from this set of curves that the resonant frequency of theresonators therefore depends essentially on the dispersions in themetallization thicknesses h.

If a sorting operation is carried out on the resonant frequenciesreducing the dispersions in the latter, the dispersions in metallizationthicknesses are thus very significantly reduced.

Moreover the applicant has established that the dispersion in staticcapacitance value depends very significantly on the dispersion inelectrode width as illustrated by FIGS. 4 and 5 relating to theevolution of the static capacitances as a function of the normalizedmetallization ratio a/p and those for the two angles of cut θ=24° andθ=34°, the set of curves being relative (θ_(min), h_(min)), (θ_(min),h_(nom)), (θ_(min), h_(max)), (θ_(nom), h_(min)), (θ_(nom), h_(nom)),(θ_(nom), h_(max)), (θ_(max), h_(min)), (θ_(max), h_(nom)), (θ_(max),h_(max)).

In parallel, the applicant was interested in the static capacitancedenoted C₀ corresponding to the capacitance created by theinter-digitated comb transducer and the successive electrodes subjectedto differences of electrical potentials. It may be shown that thedispersions in the value of this capacitance depend significantly on thedispersions in electrode widths.

Thus, a sorting operation on the values of static capacitance aimed atreducing their dispersions very significantly reduces the dispersions inelectrode widths. The static capacitance of the resonator R₁ is denotedC₀₁ and the static capacitance of the resonator R₂ is denoted C₀₂.

The principle of the present invention rests on the fact of reducing thedispersion in the difference of CTFs (1^(st) and 2^(nd) orders) withoutmeasuring the resonators individually temperature-wise.

It was demonstrated previously that f₀, C₁, C₂ depended solely on a, h,θ, and that it was possible: on the one hand to reduce the dispersion inmetallization thickness by carrying out a sorting operation on theresonant frequency and on the other hand to reduce the dispersion inelectrode width by carrying out a sorting operation on the staticcapacitance of the resonators.

It is therefore possible to reduce the dispersion in CTFs withoutmeasuring the sensors temperature-wise but by carrying out a measurementof electrical parameters at ambient temperature. However, it is notdesirable to carry out a sorting operation on the resonators separatelybut to use a matching as indicated previously so as not to penalize theyields.

An important aspect of the present invention consists therefore incarrying out a matching of R₁ and R₂ so as to reduce the dispersions inf₀₂−f₀₁ and C₀₂−C₀₁, so as ultimately to reduce the dispersions inC₁₂−C₁₁ and C₂₂−C₂₁.

A matching on f₀₂−f₀₁ and C₀₂−C₀₁ appreciably reduces the dispersions inh₂−h₁ and a₂−a₁ and the reductions in the dispersions in h₂−h₁ and a₂−a₁obtained generate an appreciable reduction in the dispersions in C₁₂−C₁₁and C₂₂−C₂₁.

Advantageously, the measurements of the electrical reflectioncoefficient S11 of the resonators are carried out with tips exhibiting acharacteristic impedance of 50 ohms and connected to a network analyser.A calibration of the tips (open circuit, short-circuit, suitable load,and correction of the phase shift related to the electrical length ofthe measurement means) will have been carried out beforehand.

A recording of the variation of the parameter S11 in the frequency bandof interest is performed. The values of the modulus and of the phase ofS11 are therefore available with a frequency sampling increment smallenough to correctly evaluate the resonant frequency (on the basis of themaximum of the conductance). A parameter fitting corresponding to thevariation of the coefficient S11 is thereafter typically performed withrespect to a model of Butterworth Van Dyck type composed of a series RLCcircuit with the static capacitance of the SAW device in parallel. Oncompletion of the fitting operation the static capacitance and theresonant frequency of the resonator at the resonant frequency aretherefore known.

An alternative scheme can also be employed; the latter consists in usinga high-precision (less than a femtoFarad) capacimeter.

The applicant has estimated the yields of a matching by aggregating theparameters f₀₂−f₀₁ and C₀₂−C₀₁ with the first series of resonators R₁and the second series of resonators R₂.

The variables f₀₁, f₀₂, C₀₁, C₀₂ are considered to be Gaussian randomvariables. The means and the standard deviations of these variables arethose arising from experimental data. It is considered for this purposethat the range is equal to 6 times the standard deviation:max(X)−min(X)=6·σ(X)

The standard deviations used are as follows:σ(f ₀₁)=σ(f ₀₂)=37 kHzσ(C ₀₁)=σ(C ₀₂)=7 fF

The algorithm used to carry out the matching does not use anyoptimization scheme, various pairs of specimens are not tested tomaximize the number of matched specimens. The set of specimens ofresonators R₁ is simply perused and for each of them a resonator R₂ isselected such that the differences f₀₂−f₀₁ and C₀₂−C₀₁ satisfy thematching criterion.

Finally, in practice, it turns out that the matching is realizable oncondition that one limits oneself to a wafer of resonator R₁ and a waferof resonator R₂ in the choice of the pairs of specimens to be matched.Now, the number of resonators that can be produced on a wafer isapproximately 1200. The calculated yields therefore correspond to amatching of 1200 specimens of resonators R₁ and 1200 specimens ofresonators R₂.

Table 1 below presents the values of the yields achievable as a functionof the matching criterion:

Matching criterion Matching Matching based on f₀₂ − f₀₁ and criterionbased on criterion based on C₀₂ − C₀₁ f₀₂ − f₀₁ in kHz C₀₂ − C₀₁ in fFYield in % +/− σ(X) +/−37 +/−7 99.25 +/−0.5 σ(X) +/−17.5 +/−3.5 97.6+/−0.2 σ(X) +/−7.4 +/−1.4 87.7 +/−0.1 σ(X) +/−3.7 +/−0.7 71.6 +/−0.05σ(X) +/−1.85 +/−0.35 47.1 +/−0.01 σ(X) +/−0.37 +/−0.07 3.5 +/−0.005 σ(X)+/−0.185 +/−0.035 1.3

The two cases of matching to +/−0.2 σ(X) and +/−0.1 σ(X) areparticularly interesting in so far as they lead to yields ofrespectively 87.7% and 71.6%, which are compatible with industrialobjectives and impose attainable constraints in terms of dispersion.

Indeed, in each case, the dispersion in a₂−a₁ is calculated first of allon the basis of the dispersion in C₀₂−C₀₁ by considering that C₀₂−C₀₁depends solely on a₂−a₁. The uncertainty in f₀₂−f₀₁ is then calculatedon the basis of the calculated dispersion in a₂−a₁ and of the dispersionin θ₂−θ₁, and this is added to the matching criterion based on f₀₂−f₀₁to get the total span of variations of f₀₂−f₀₁ that is attributable toh₂−h₁ (allowance for the case where the variations due to h₂−h₁ andthose due to a₂−a₁ and θ₂−θ₁ are of opposite signs). Having calculatedthe total span of variations of f₀₂−f₀₁ that is attributable to h₂−h₁,the dispersion in h₂−h₁ is calculated. Finally, knowing the dispersionsin h₂−h₁, a₂−a₁ and θ₂−θ₁, the dispersions in C₁₂−C₁₁ and C₂₂−C₂₁ arecalculated.

The results associated with the 2 cases, as well as the intermediatesteps, are summarized in Table 2 below.

Matching +/−0.2 σ(X) +/−0.1 σ(X) Criterion (+/−7.4 kHz / +/−1.4 fF)(+/−3.7 kHz / +/−0.7 fF)$\frac{\Delta\left( {a_{2} - a_{1}} \right)}{p}$ +/−0.0018 +/−0.0012Uncertainty   +/−5.5 kHz   +/−5.5 kHz f₀₂ − f₀₁ due to Δ(θ₂ − θ₁)Uncertainty  +/−1.25 kHz  +/−1.15 kHz f₀₂ − f₀₁ due to Δ(a₂ − a₁) Total+/−14.15 kHz +/−10.35 kHz Uncertainty$\frac{\Delta\left( {h_{2} - h_{1}} \right)}{p}$ +/−0.0068% +/− 0.0054%Order of Differential Temperature Coefficiencs C₁ C₂ C₁ C₂ Δ(C₂ − C₁)due +/−0.0305 +/−0.035 +/−0.0285 +/−0.0305 to Δ(h₂ − h₁) ppm/° C. ppb/°C². ppm/° C. ppb/° C². Δ(C₂ − C₁) due +/−0.0315 +/−0.0315 +/−0.027+/−0.0305 to Δ(a₂ − a₁) ppm/° C. ppb/° C². ppm/° C. ppb/° C². Δ(C₂ − C₁)due +/−0.045 +/−0.053 +/−0.045 +/−0.053 to Δ(θ₂ − θ₁) ppm/° C. ppb/° C².ppm/° C. ppb/° C². Sum of +/−0.107 +/−0.12 +/−0.101 +/−0.114Differential ppm/° C. ppb/° C². ppm/° C. ppb/° C². CoefficientsDispersions

On the basis of the previously calculated dispersions (last line oftable 2), it is possible to determine the reduction in the error in themeasurement of the temperature obtained.

For this purpose, first of all the mean calibration coefficients arecalculated on the basis of the mean parameters (f₀, C₁, C₂) obtained bysimulation for each resonator.

Next, random draws are carried out on the basis of the dispersionsobtained.

For f₀₂−f₀₁, a uniform distribution in [−Δ(f₀₂−f₀₁), Δ(f₀₂−f₀₁)] is usedsince f₀₂−f₀₁ is matched directly and since the matching criterion issmall compared with the range of the initial Gaussian. For C₁₂−C₁₁ andC₂₂−C₂₁, a Gaussian distribution is used based on the dispersionscalculated previously (Δ(X)=3·σ(X)). More precisely, we calculate:3·σ(s)=Δs≈f ₀₁·Δ(C ₁₂ −C ₁₁)3·σ(ε)=Δε≈f ₀₁·Δ(C ₂₂ −C ₂₁)

Next, Gaussian random draws of s with standard deviation σ(s) and of εwith standard deviation σ(ε) are carried out.

The parameters (mean values) used are as follows:

E[C₁₁]=6.8 ppm/° C.

E[C₂₁]=−30.7 ppb/° C.²

E[C₁₂]=0.4 ppm/° C.

E[C₂₂]=−38.1 ppb/° C.²

E[f₀₁]˜433.4 MHz

E[f₀₂]˜434.5 MHz

The temperature span considered by way of example is defined by Tε[−20,250]° C.

1) For a matching to +/−0.2 σ(X):Δ(f ₀₂ −f ₀₁)=7.4 kHzσ(s)=0.036 ppm/° C.*433.4 MHz=15.6 Hz/Cσ(ε)=0.04 ppb/° C.²*433.4 MHz=0.0173 Hz/C²

We obtain:3·σ(Err)=5.75° C. and 99.74% of the population in the interval[−3.62,3.62]° C.

2) For a matching to +/−0.1 σ(X)Δ(f ₀₂ −f ₀₁)=3.7 kHzσ(s)=0.034 ppm/° C.*433.4 MHz=14.735 Hz/Cσ(ε)=0.038 ppb/° C.²*433.4 MHz=0.0165 Hz/C²

We obtain:3·σ(Err)=3.55° C. and 99.74% of the population in the interval[−2.81,2.81]° C.

FIGS. 6 and 7 show that a matching operation for the criterion +/−0.2σ(X) leads to the obtaining of a calibration-free temperature sensoroperating in the span −20° C. to 250° C. exhibiting a precision of+/−3.6° C. throughout the span with a matching yield of 87.7% and that amatching operation for the criterion +/−0.1σ(X) generates a decrease inthe yield (71.6%) but makes it possible to obtain a calibration-freesensor with a better precision (+/−2.8° C.) in the same temperaturespan.

The many features and advantages of the invention are apparent from thedetailed specification, and, thus, it is intended by the appended claimsto cover all such features and advantages of the invention which fallwithin the true spirit and scope of the invention. Further, sincenumerous modifications and variations will readily occur to thoseskilled in the art, it is not desired to limit the invention to theexact construction and operation illustrated and described, and,accordingly, all suitable modifications and equivalents may be resortedto that fall within the scope of the invention.

1. A method of collective fabrication of remotely interrogatable sensors, each sensor includes at least one first resonator and at least one second resonator, each of the at least one resonator includes acoustic wave transducers that exhibit a first and a second operating frequency, the method comprising: fabricating a first series of the at least one first resonators exhibiting a first resonant frequency at ambient temperature and a first static capacitance; fabricating a second series of the at least one second resonators exhibiting a second resonant frequency at ambient temperature and a second static capacitance; performing a series of electrical measurements of the first series of the at least one first resonators and the second series of the at least one second resonators to determine first pairs and second pairs of resonant frequency and of capacitance of each of the at least one first resonators and the at least one second resonators; and performing a series of matching of a first resonator and of a second resonator based at least in part on the aggregate of two criteria: (a). a dispersion in a difference in resonant frequency of each of the at least one first resonators and the at least one second resonators is less than a first threshold value; and (b) a dispersion in a difference in capacitance of each of the at least one first resonators and the at least one second resonators is less than a second threshold value.
 2. The method according to claim 1, wherein the series of electrical measurements are performed by measurement of a reflection coefficient S11 or measurement of an admittance Y11 or measurement of an impedance Z11.
 3. The method according to claim 1, wherein the series of electrical measurements are performed with a network analyzer.
 4. The method according to claim 1, wherein the series of electrical measurements of capacitance are performed with a capacimeter.
 5. The method according to claim 1, wherein the first pairs and the second pairs of the resonant frequencies are similar and situated in ISM frequency span from 433.05 MHz to 434.79 MHz, and the first threshold value is less than or equal to about a kiloHertz.
 6. The method according to claim 1, wherein the first pairs and the second pairs of the resonant frequencies are similar and situated in the ISM frequency span from 433.05 MHz to 434.79 MHz, and the second threshold value is less than a femtoFarads.
 7. The method according to claim 1, wherein for each of the first resonator of the first series of the at least one first resonators and a selection of a second resonator of the second series of the at least one second resonators satisfies the two criteria.
 8. The method according to claim 1, wherein the first series of the at least one first resonators are fabricated on a first substrate and the second series of the at least one second resonators are fabricated on a second substrate.
 9. The method according to claim 1, wherein the first series of the at least one first resonators and the second series of the at least one second resonators are produced on first and second quartz substrates of different cuts.
 10. The method according to claim 9, wherein the first and second quartz substrates are defined by angles of cut Θ, of 24° and 34° in order to generate frequencies of 433 MHz and 434 MHz.
 11. The method according to claim 1, further comprising: fabricating the first series of the at least one first resonators on a first substrate and fabricating the second series of the at least one second resonators on a second substrate; performing unit slicing of first and of second chips including the first series of the at least one first resonators and the second series of the at least one second resonators, respectively, from the first and second substrates; matching the first and second chips; and assembling of the first and second chips in a package.
 12. The method according to claim 1, further comprising: fabricating the first series of the at least one first resonators on a first substrate and fabricating the second series of the at least one second resonators on a second substrate; performing unit slicing of first and of second chips including the first series of the at least one first resonators and the second series of the at least one second resonators, respectively, from the first and second substrates; performing an individual packaging of the first and second chips in individual packages; matching a first and of a second chips from a previously package.
 13. The method according to claim 1, wherein the sensor is a temperature sensor.
 14. The method according to claim 1, wherein the first series of the at least one first resonators are oriented on a first substrate in a first direction, the second series of the at least one second resonators are oriented on a second substrate in a second direction, the first and second directions corresponding to directions of propagation of surface waves, and the first direction makes a non-zero angle with the second direction. 